Introduction
============


.. _Ref194217701:




Why use X- FEM?
-------------------------

The X- FEM [:ref:`1 <1>`] method is a simple extension of the finite element method. It allows a mesh independent of the geometry of the problem. Borders, holes, cracks become entities that can be inserted, moved, propagated, without having to modify the mesh. A simple and unique mesh thus replaces several complex meshes. To represent a discontinuity or a singularity within finite elements, the base of form functions is enriched, using the properties of the unit partition. In cracking, the displacement (or temperature) discontinuity due to the crack is introduced by a generalized Heaviside function and the addition of asymptotic fields at the crack end improves the precision in elastic fracture mechanics. In addition, the "level sets" method is particularly practical for representing 3D cracks and effective for the propagation phase, the main idea being to consider the interface as the iso-zero of a distance function.

You can use X- FEM to:


* represent a crack (discontinuity of movements or temperature),


* represent an interface between two disjoint solids (discontinuity of movements or temperature),


* represent a hole or a void (underthickness for example),


* represent the interface between two materials (stress discontinuities).

*Representation of a crack*

In fracture mechanics, a crack has two lips and a crack bottom. The lips are initially confused, otherwise it is called a cut. The field of movement (or temperature) is discontinuous across the lips of the crack, and the stress field is singular at the bottom of the crack.

In *Code_Aster*, it is possible to define a crack (see the implementation in § :ref:`18 <RefNumPara__29223511>`) using two level set functions and to have it propagated.

*Representation of an interface between two disjoint solids*

The two solids are separated by an interface (examples: a geological fault between two layers, the crushing of a piece of land on a fixed structure, two thermally insulated solids). The displacement or temperature field is discontinuous across the interface. The materials on either side of the interface may be different, but have the same law of behavior. Contact is possibly defined on the interface.

This functionality is possible in*Code_Aster*, by defining the interface with a level set function (see the implementation in § :ref:`4 <Ref194209080>`).

*Representation of a hole or a vacuum*

This is a particular case of the previous case where one of the solids is empty. For this, no contact conditions are defined on the interface. If no load is applied to the solid corresponding to the vacuum (except the blocking of rigid modes) it is not involved in the calculation, and plays the role of "vacuum".

This functionality is possible in *Code_Aster*, by defining the interface between matter and vacuum by a level set function (see the implementation in § :ref:`5 <Ref194214607>`).

*Representation of an interface in a bimaterial*

In a bi-material, the two materials are "glued", the movement across the interface between the two materials is continuous, but the stresses are discontinuous.

This feature is not possible in*Code_Aster*.

In the rest of this document, the general term "crack" will be used, which will refer to either a crack or an interface.




Specificity of a calculation with X- FEM
----------------------------------

Compared to a classical calculation, some steps are specific:


* creation of the mesh: the mesh does not have a crack,


* definition of the crack: since the crack is not contained in the mesh, it must be defined in another way,


* modification of the model: some elements must be enriched in order to represent a discontinuity of movement or temperature through the crack and the singularity of stress at the bottom of the crack,


* visualization post-processing: in order to visualize the opening of the crack or the discontinuity of the temperature field, it is necessary to create a visualization mesh and the associated result fields.




Related documentation
----------------------

User Documentation:


* DEFI_FISS_XFEM [U4.82.08];


* MODI_MODELE_XFEM [U4.44.11];


* POST_MAIL_XFEM [:external:ref:`U4.82.21 <U4.82.21>`];


* POST_CHAM_XFEM [:external:ref:`U4.82.22 <U4.82.22>`];


* PROPA_FISS [U4.82.11].

Validation documentation (test cases):


* FORMA01: Practical work from the "Initiation" training: Plate with holes in elasticity: the D modeling is treated with X- FEM [:external:ref:`V3.02.326 <V3.02.326>`];


* FORMA06: Practical work from the "Advanced Use" training: multi-cracked plate under traction [:external:ref:`V3.02.112 <V3.02.112>`];


* FORMA07: Practical work from the "Advanced Use" training: circular crack in an infinite environment (B modeling is treated with X- FEM) [:external:ref:`V3.04.156 <V3.04.156>`];


* HPLV103: Calculation of :math:`\mathit{KI}` and :math:`G` 3D thermo-elastic for a circular crack [:external:ref:`V7.03.103 <V7.03.103>`] - Validation of DEFI_FOND_FISS;


* HSNV132: Fissure X- FEM in thermo-elasticity [:external:ref:`V7.22.132 <V7.22.132>`];


* HSLA304: Thermo-mechanical chaining with X- FEM, restricted access [:external:ref:`V7.12.304 <V7.12.304>`];


* SDLS114: Calculation of the stress intensity factors of a cracked plate by modal recombination [:external:ref:`V2.03.114 <V2.03.114>`];


* SDLS120: Cracked 2D plate subjected to a loading in Mode :math:`I`. Validating the modal calculation with X- FEM [:external:ref:`V2.03.120 <V2.03.120>`];


* SDLV130: Cracked 3D plate subjected to a loading in Mode :math:`I`. Validating the modal calculation with X- FEM [:external:ref:`V2.03.130 <V2.03.130>`];


* SSLP313 (DEFI_FOND_FISS): Inclined crack in an unlimited plate, subject to uniform traction to infinity [:external:ref:`V3.02.313 <V3.02.313>`] - Validation of DEFI_FOND_FISS;


* SSLP315: Propagation of an inclined crack in a 2D plate [:external:ref:`V3.02.315 <V3.02.315>`];


* SSLP316: Validation of the error estimator X- FEM on a cracked plate [:external:ref:`V3.02.316 <V3.02.316>`];


* SSLP317: Validation of the RAFF_XFEM macro-command on a multi-cracked plate [:external:ref:`V3.04.317 <V3.04.317>`];


* SSLP318: Propagation of a non-through X-FEM crack solicited in :math:`I` [:external:ref:`V3.02.318 <V3.02.318>`] mode;


* SSLP319: Propagation of two unblocking XFEM cracks stressed in :math:`I` [:external:ref:`V3.02.319 <V3.02.319>`] mode;


* SSLP320: Propagation of an open X-FEM crack solicited in Mode :math:`I` [:external:ref:`V3.02.320 <V3.02.320>`];


* SSLP321: Propagation of an X- FEM crack in a 3-point flexure plate [:external:ref:`V3.02.321 <V3.02.321>`];


* SSLP322: Propagation of an X- FEM crack in a 3-point flexure plate with 3 holes [:external:ref:`V3.02.322 <V3.02.322>`];


* SSLP323: Propagation of a radial crack leading to a rotating disk [:external:ref:`V3.02.323 <V3.02.323>`];


* SSLV110: Elliptical crack in an infinite medium [:external:ref:`V3.04.110 <V3.04.110>`];


* SSLV134: Circular crack in an infinite medium [:external:ref:`V3.04.134 <V3.04.134>`];


* SSLV154: Circular crack in mixed mode [:external:ref:`V3.04.154 <V3.04.154>`];


* SSLV155: Lens crack under traction [:external:ref:`V3.04.155 <V3.04.155>`];


* SSLV311: Crack in a quarter ellipse at the corner of a thick rotating disk [:external:ref:`V3.04.311 <V3.04.311>`] - Validation of DEFI_FOND_FISS;


* SSLV313: Cracked pressure pipe - validation of method X- FEM [:external:ref:`V3.04.313 <V3.04.313>`];


* SSLV314: Propagation of a plane crack in pure I mode in 3D [:external:ref:`V3.04.314 <V3.04.314>`];


* SSLV315: Propagation of an inclined crack in mixed mode in 3D [:external:ref:`V3.04.315 <V3.04.315>`];


* SSLV316: Propagation cracking imposed with X- FEM [:external:ref:`V3.04.316 <V3.04.316>`];


* SSLV319: Plane crack of a semi-elliptical crack [V3.04.319];


* SSLV320: Plane propagation of a 3D crack dividing and merging with X- FEM [:external:ref:`V3.04.320 <V3.04.320>`];


* SSLV321: Validation of pre-conditioning X- FEM for a grazing interface [:external:ref:`V3.04.321 <V3.04.321>`];


* SSNP110: Edge crack in a rectangular plate finished in elasto-plasticity [:external:ref:`V6.03.110 <V6.03.110>`];


* SSNP118: Validation of joint and interface elements in 2D plane and 3D [:external:ref:`V6.03.118 <V6.03.118>`];


* SSNP133: Cracking a perforated plate with cohesive models [:external:ref:`V6.03.133 <V6.03.133>`];


* SSNP138: 2D angled crack with X- FEM [:external:ref:`V6.03.138 <V6.03.138>`];


* SSNP144: Use of a cohesive zone model with the X- FEM method [:external:ref:`V6.03.144 <V6.03.144>`];


* SSNP503: Contact in large slides with X- FEM for horizontal cracks [:external:ref:`V6.03.503 <V6.03.503>`];


* SSNP504: Contact in large slides with X- FEM for oblique cracks [:external:ref:`V6.03.504 <V6.03.504>`];


* SSNP505: Multi-cracked bitraction/shear plate with X- FEM [:external:ref:`V6.03.505 <V6.03.505>`];


* SSNV108: European CT - Round Robin specimen in Fracture Mechanics (1985) [:external:ref:`V6.04.108 <V6.04.108>`];


* SSNV166: Cracked cylinder under multiple loads [:external:ref:`V6.04.166 <V6.04.166>`] - Validation of DEFI_FOND_FISS;


* SSNV173: Cracked bar with X- FEM [:external:ref:`V6.04.173 <V6.04.173>`];


* SSNV182: Block with interface in frictional contact with X- FEM [:external:ref:`V6.04.182 <V6.04.182>`];


* SSNV185: An opening crack in a 3D plate of width finished with X- FEM [:external:ref:`V6.04.185 <V6.04.185>`];


* SSNV186: LBB condition and rubbing contact with X- FEM [:external:ref:`V6.04.186 <V6.04.186>`];


* SSNV191: Validation of Neumann conditions with X- FEM in 3D [:external:ref:`V6.04.191 <V6.04.191>`];


* SSNV192: Test specimen with central crack X- FEM [:external:ref:`V6.04.192 <V6.04.192>`];


* SSNV195: Multi-cracked bar with X- FEM [:external:ref:`V6.04.195 <V6.04.195>`];


* SSNV198: Dirichlet conditions with X- FEM in 3D [:external:ref:`V6.04.198 <V6.04.198>`];


* SSNV201: Block with interface in sliding contact with X- FEM [:external:ref:`V6.04.201 <V6.04.201>`];


* SSNV203: Applying pressure to the lips of a crack with X- FEM [:external:ref:`V6.04.203 <V6.04.203>`];


* SSNV209: Contact/rub interface with X- FEM [:external:ref:`V6.04.209 <V6.04.209>`];


* SSNV245: Imposition of Dirichlet conditions on heaviside X-FEM elements using a space function [:external:ref:`V6.04.245 <V6.04.245>`];


* SSNV246: Application of distributed pressure on the lips of a crack X- FEM curve crossing a column [:external:ref:`V6.04.246 <V6.04.246>`];


* SSNV247: Application of distributed pressure on the lips of a curved X-FEM crack crossing a spherical cap [:external:ref:`V6.04.247 <V6.04.247>`];


* SSNV507: Rotating a rigid inclusion with X- FEM [:external:ref:`V6.04.507 <V6.04.507>`];


* SSNV508: Block with traction interface and lateral pressure without contact [:external:ref:`V6.04.508 <V6.04.508>`];


* SSNV509: Chain in contact rubbing with X-FEM quadratic [:external:ref:`V6.04.509 <V6.04.509>`];


* SSNV510: Uniaxial compression of a multi-cracked block [:external:ref:`V6.04.510 <V6.04.510>`];


* SSNV511: Block cut by two interfaces intersected with X- FEM [:external:ref:`V6.04.511 <V6.04.511>`];


* SSNV512: Block cut by a vertical crack connecting between two horizontal cracks with X- FEM [:external:ref:`V6.04.512 <V6.04.512>`];


* SSNV513: Block cut by three interfaces that connect sequentially with X- FEM [:external:ref:`V6.04.513 <V6.04.513>`];


* TTLA203: Cylinder with adiabatic crack at imposed temperatures [:external:ref:`V4.19.203 <V4.19.203>`];


* TTLP101: Plate cracked at imposed temperatures with the condition of exchange through the lips of the crack [:external:ref:`V4.23.101 <V4.23.101>`]


* TPLP305: Bar at fixed temperatures with an adiabtic interface of type X- FEM [:external:ref:`V4.05.305 <V4.05.305>`];


* WTNV143: Application of distributed pressure on the lips of a crack X- FEM for the hydromechanical case [:external:ref:`V7.31.143 <V7.31.143>`];


* ZZZZ255: Validating option TEST_MAIL in PROPA_FISS [V1.01.255];


* ZZZZ282: Validation of the definition of a crack on a grid by DEFI_FISS_XFEM [V1.01.282];


* ZZZZ283: Validation of the use of a grid with an X- FEM crack on a mesh refined by Homard [:external:ref:`V1.01.283 <V1.01.283>`];


* ZZZZ298: POST_K1_K2_K3 computer validation [V1.01.298];


* ZZZZ309: Validation of topological options ZONE_MAJet TOREdans DEFI_GROUP [V1.01.309];


* ZZZZ339: Validation of the G calculation with initial constraints in 3D [:external:ref:`V1.01.339 <V1.01.339>`];


* ZZZZ340: Validating the TEMP_CONTNUE keyword for AFFE_CHAR_THER [V1.01.340];


* ZZZZ346: Bifurcation of a straight XFEM interface with quadratic 2D elements [V1.01.346];


* ZZZZ358: Validation of the MODELE_THER keyword for the MODI_MODELE_XFEM operator [V1.01.358];


* ZZZZ361: Degenerate cutting cases for quadratic 3D X- FEM elements [V1.01.361];


* ZZZZ362: Checking the consistency of the fields produced by the sub-division X- FEM [:external:ref:`V1.01.362 <V1.01.362>`];

Reference documentation:


* Extended Finite Element Method: General [:ref:`R7.02.12 <R7.02.12>`];


* Crack propagation algorithms with X- FEM [:ref:`R7.02.13 <R7.02.13>`];


* Contact in small slips with X- FEM [:ref:`R5.03.54 <R5.03.54>`];


* Contact in large swings with X- FEM [:external:ref:`R5.03.53 <R5.03.53>`].


* Hydromechanical coupling with XFEM [:ref:`R7.0.18 <R7.0.18>`].

Computer description:


* Data structures linked to X- FEM [:ref:`D4.10.02 <D4.10.02>`];


* Implementation of the "major changes with X- FEM" approach [:ref:`D9.05.06 <D9.05.06>`].